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2.3 Probability applications of counting principles
We are now consider some complicate sample space and events. To count the number of outcomes of
sample space and events, we will use multiplication principle, permutation and combination. Recall the
probability of the event E is
n(E)
.
P (E) =
n(S)
Example 1. A fair coin is tossed six times. Assuming that any outcome is as likely as any other, find
the probability of obtaining exactly three heads.
P (E) =
C(6, 3)
26
Example 2. Find the probability of drawing a flush (5 cards in a single suit: S,H,C,D).
P (E) =
4 × C(13, 5)
C(52, 5)
Example 3. A bag contains 5 green marbles, 3 blue marbles, and 7 purple marbles. A sample of five
marbles is selected.
a) What is the probability that the sample of 5 marbles consists of exactly 2 green marbles and 3 purple
marbles?
C(5, 2)C(7, 3)
C(15, 5)
b) What is the probability that the sample of 5 marbles consists of at most 1 green marbles?
C(10, 5) + C(5, 1)C(10, 4)
C(15, 5)
c) What is the probability that the sample of 5 marbles consist of exactly 2 green marbles or exactly 3
purple marbles?
C(5, 2)C(10, 3) + C(7, 3)C(8, 2) − C(5, 2)C(7, 3)
C(15, 5)
Example 4. Suppose we have a jar with 8 blue and 6 green marbles. Find the probability distribution
table for the number of blue marbles in the sample of 2 marbles?
Outcome
Probability
0 blue marbles exactly 1 blue marbles
C(6,2)
C(14,2)
C(8,1)C(6,1)
C(14,2)
exactly 2 blue marbles
C(82)
C(14,2)
Example 5. Suppose there are 30 students in class today. What is the probability that at least 2 of
these people will have the same birthday (assume 365 days for a year)?
36530 − P (365, 30)
36530
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2.4 Bernoulli Trials
Bernoulli Trials: A experiment with just two outcomes.
Example 6.
• Flip a coin and see if heads or tails turns up.
• Test a transistor to see if it is defective or not.
• Examine a patient to if a particular disease is present or not.
• Take a free throw in basketball and make the basket or not.
Notation: We commonly refer to the two outcomes of a Bernoulli trial as “success” (S) or “failure”
(F).
Note: if p is the probability of “success”, q is the probability of “failure”, p + q = 1.
We are interested in conducting Bernoulli trials many times. We call repeated Bernoulli trial. For
example, flip a coin six times. We know that the number of the sample space is 26 . When we consider
probabilities of events in repeated Bernoulli trials, we make the following fundamental assumption.
Fundamental assumption for Bernoulli trials: Successive Bernoulli trials are independent of one
another.
Example 7. You flip a fair coin 12 times. Find the probability of flipping exactly 5 heads.
12
1
C(12, 5)
2
Example 8. A student takes a four-question single-choice test by guessing. Each question has four
possible answers. What is the probability that he or she answers exact 2 questions correctly?
2 2
1
3
C(4, 2)
4
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General setting: Given a Bernoulli trial repeated n times, we want to find the probability that
a specific number of successes occur. Notation: P (X = r). Let the probability of “success” be p.
So
P (X = r) = C(n, r)pr (1 − p)n−r
To compute the above formula in calculator:
1. Press 2nd.
2. Press DISTR (above VARS key).
3. Select binompdf( (Option A).
4. Enter the number of trials, n; the comma key, the probability of success, p; the comma key, and
then the number of successful trials, r.
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5. Press ENTER.
Example 9. Find the probability of getting exactly 3 heads if you toss a coin 7 times that has been
shown to result in tails with probability 0.6.
To compute P (X ≤ r) in calculator:
1. Press 2nd.
2. Press DISTR (above VARS key).
3. Select binomcdf( (Option B).
4. Enter the number of trials, n; the comma key, the probability of success, p; the comma key, and
then the number of successful trials, r.
5. Press ENTER.
Example 10. The manager of a movie rental store knows that 40% of the people who are browsing in
the store will actually rent a movie. What is the probablilty that among 20 people who are browsing in
the store
a) At most five will rent a movie? (Round your answer to 5 decimal places.) P (X ≤ 5)
b) At least five will rent a movie? (Round your answer to 5 decimal places.) P (X ≥ 5) = 1 − P (X ≤ 4)
c) More than 3 but fewer than 10 will rent a movie? (Round your answer to 5 decimal places.)
P (3 < X < 10) = P (X ≤ 9) − P (X ≤ 3)
Example 11. The probability that a DVD player produced by a company is defective is estimated to
be 0.09. If a sample of 14 DVD players is selected at random, what is the probabililty that the sample
contains between 2 and 10 defectives, inclusive. (Round your answers to four decimal places.)
P (2 ≤ X ≤ 10) = P (X ≤ 10) − P (X ≤ 1)
3.1 Random Variable
A random variable is a rule that assigns precisely one real number to each outcome of an experiment.
We usually denote a random variable by X.
Notes:
• The assignments of numbers to the outcomes of experiments are normally done in a manner that
is reasonable and, most importantly, in a manner that permits these numbers to be used for
interpretation and comparison.
• Unless otherwise specified, when the outcomes of an experiment are themselves numbers, the
random variable is the rule that assigns each number to itself.
Example: Probability distribution.
Types of Random Variables:
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1. A random variable is finite discrete if it assumes only a finite number of values.
2. A random variable is infinite descrete if it takes on an infinite number of values that can be listed
in a sequence, so that there is a first one, a second one, a third one, and so on.
3. A random variable is said to be continuous number of values in some interval of real numbers.
Example 12. a) Flip a fair coin. Assign the random variables 1 to head and 2 to tail.
b) Flip a fair coin until a head is obtained. Assign the random variable X to number of flips until a
head is shown.
c) Measure the height of adult men in this country. Assign the random variable X to the height measured
in feet.
Example 13. Three marbles are selected at random without replacement from a bag containing 4 green
marbles and 6 purple marbles. Let the random variable X denote the number of purple marbles drawn.
Find P (X = 2) and P (X ≥ 2).
Example 14. An unfair die has the property that the probability of rolling a 1 is 0.3. This die is rolled
4 times. Let X be the number of times a 1 is rolled. Find the probability distribution for X. Round the
probabilities to 4 decimal places.
Histograms
The area and probability of a region of a histogram associated with the random variable X is equal to
P (X), the probability that X occurs. Furthermore, the probability that X takes on the values in the
range Xi ≤ X ≤ Xj is the sum of the areas of the histogram from Xi to Xj .
Example 15. You are dealt a hand of three cards. Find the probability distribution for the number of
clubs. Graph this in a histogram.
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